Interpolants for Runge-Kutta formulas
ACM Transactions on Mathematical Software (TOMS)
Natural continuous extensions of Runge-Kutta formulas
Mathematics of Computation
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Parallel solution of ODE's by multi-block methods
SIAM Journal on Scientific and Statistical Computing
Stability properties of interpolants for Runge-Kutta methods
SIAM Journal on Numerical Analysis
Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
A-stable parallel block methods for ordinary and integro-differential equations
Selected papers from the international conference on Numerical solution of Volterra and delay equations
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
L-stable parallel one-block methods for ordinary differential equations
SIAM Journal on Numerical Analysis
Some applications of continuous Runge-Kutta methods
Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
Stability of implicit one-block methods for delay differential equations
Applied Numerical Mathematics
Continuous block implicit hybrid one-step methods for ordinary and delay differential equations
Applied Numerical Mathematics
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Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations (ODEs), delay differential equations, neutral differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ODEs generated by a class of block $\theta$-methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous block $\theta$-methods for ODEs are studied. As an application, we adopt the continuous block $\theta$-methods to solve delay differential equations and prove that the continuous block $\theta$-methods are $GP$-stable if and only if they are $A_{\omega}$-stable for ODEs. Several numerical experiments are given to illustrate the performance of the continuous block $\theta$-methods.